zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

25sin(x)cos(x)-5sin(x)+20cos(x)=4

解答

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

解答

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

+1

度数

x = 78.46304 \dots^{\circ} + 36 0^{\circ} n, x = 281.53695 \dots^{\circ} + 36 0^{\circ} n, x = - 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 233.13010 \dots^{\circ} + 36 0^{\circ} n

求解步骤

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

两边减去

4

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 = 0

分解

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 : (5 cos (x) - 1) (5 sin (x) + 4)

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4

= (25 sin (x) cos (x) - 5 sin (x)) + (20 cos (x) - 4)

从

20 cos (x) - 4

分解出因式

4

:

4 (5 cos (x) - 1)

20 cos (x) - 4

将

20

改写为

5 \cdot 4

= 5 \cdot 4 cos (x) - 4

因式分解出通项

4

= 4 (5 cos (x) - 1)

从

25 sin (x) cos (x) - 5 sin (x)

分解出因式

5 sin (x)

:

5 sin (x) (5 cos (x) - 1)

25 sin (x) cos (x) - 5 sin (x)

将

25

改写为

5 \cdot 5

= 5 \cdot 5 sin (x) cos (x) - 5 sin (x)

因式分解出通项

5 sin (x)

= 5 sin (x) (5 cos (x) - 1)

= 4 (5 cos (x) - 1) + 5 sin (x) (5 cos (x) - 1)

因式分解出通项

5 cos (x) - 1

= (5 cos (x) - 1) (5 sin (x) + 4)

(5 cos (x) - 1) (5 sin (x) + 4) = 0

分别求解每个部分

5 cos (x) - 1 = 0 or 5 sin (x) + 4 = 0

5 cos (x) - 1 = 0 : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 cos (x) - 1 = 0

将

1

到右边

5 cos (x) - 1 = 0

两边加上

1

5 cos (x) - 1 + 1 = 0 + 1

化简

5 cos (x) = 1

5 cos (x) = 1

两边除以

5

5 cos (x) = 1

两边除以

5

\frac{5 cos ( x )}{5} = \frac{1}{5}

化简

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

使用反三角函数性质

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 sin (x) + 4 = 0 : x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

5 sin (x) + 4 = 0

将

4

到右边

5 sin (x) + 4 = 0

两边减去

4

5 sin (x) + 4 - 4 = 0 - 4

化简

5 sin (x) = - 4

5 sin (x) = - 4

两边除以

5

5 sin (x) = - 4

两边除以

5

\frac{5 sin ( x )}{5} = \frac{- 4}{5}

化简

sin (x) = - \frac{4}{5}

sin (x) = - \frac{4}{5}

使用反三角函数性质

sin (x) = - \frac{4}{5}

sin (x) = - \frac{4}{5}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

合并所有解

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn, x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

以小数形式表示解

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

作图

流行的例子

sin (θ) = - 0.9

cos(sqrt(2)T)=1,0<= T<= 2pi

cos (2 T) = 1, 0 \leq T \leq 2 π

2-2cos^2(x/2)=2cos^2(x)

2 - 2 cos^{2} (\frac{x}{2}) = 2 cos^{2} (x)

sec^2(θ)-2=tan^2(θ)

sec^{2} (θ) - 2 = tan^{2} (θ)

cos(x)=sin(x)+1

cos (x) = sin (x) + 1